Derived algebraic geometry is a branch of mathematics that generalizes
algebraic geometry to a situation where
commutative rings, which provide local charts, are replaced by either
differential graded algebras (over
),
simplicial commutative rings or
-ring spectra from
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's
scheme theory
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different s ...
allows the structure sheaf to carry
nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for
intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
(or
motivic homotopy theory
In music, a motif IPA: ( /moʊˈtiːf/) (also motive) is a short musical phrase, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition: "The moti ...
) of singular algebraic varieties and
cotangent complex In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic ob ...
es in
deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesi ...
(cf. J. Francis), among the other applications.
Introduction
Basic objects of study in the field are
derived schemes and
derived stacks. The oft-cited motivation is
Serre's intersection formula In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection numb ...
. In the usual formulation, the formula involves the
Tor functor and thus, unless higher Tor vanish, the
scheme-theoretic intersection (i.e., fiber product of immersions) ''does not'' yield the correct
intersection number. In the derived context, one takes the
derived tensor product In algebra, given a differential graded algebra ''A'' over a commutative ring ''R'', the derived tensor product functor is
:- \otimes_A^ - : D(\mathsf_A) \times D(_A \mathsf) \to D(_R \mathsf)
where \mathsf_A and _A \mathsf are the categories of ri ...
, whose higher homotopy is higher Tor, whose
Spec is not a scheme but a
derived scheme. Hence, the "derived" fiber product yields the correct intersection number. (Currently this is hypothetical; the derived intersection theory has yet to be developed.)
The term "derived" is used in the same way as
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in var ...
or
derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pro ...
, in the sense that the category of commutative rings is being replaced with a
∞-category of "derived rings." In classical algebraic geometry, the derived category of
quasi-coherent sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
is viewed as a
triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cate ...
, but it has natural enhancement to a
stable ∞-category In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that
*(i) It has a zero object.
*(ii) Every morphism in it admits a fiber and cofiber.
*(iii) A triangle in it is a fiber sequence if and only if it is a co ...
, which can be thought of as the
∞-categorical analogue of an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
.
Definitions
Derived algebraic geometry is fundamentally the study of geometric objects using homological algebra and homotopy. Since objects in this field should encode the homological and homotopy information, there are various notions of what derived spaces encapsulate. The basic objects of study in derived algebraic geometry are derived schemes, and more generally, derived stacks. Heuristically, derived schemes should be functors from some category of derived rings to the category of sets
:
which can be generalized further to have targets of higher groupoids (which are expected to be modelled by homotopy types). These derived stacks are suitable functors of the form
:
Many authors model such functors as functors with values in simplicial sets, since they model homotopy types and are well-studied. Differing definitions on these derived spaces depend on a choice of what the derived rings are, and what the homotopy types should look like. Some examples of derived rings include commutative differential graded algebras, simplicial rings, and
-rings.
Derived geometry over characteristic 0
Over characteristic 0 many of the derived geometries agree since the derived rings are the same.
algebras are just commutative differential graded algebras over characteristic zero. We can then define derived schemes similarly to schemes in algebraic geometry. Similar to algebraic geometry, we could also view these objects as a pair
which is a topological space
with a sheaf of commutative differential graded algebras. Sometimes authors take the convention that these are negatively graded, so
for
. The sheaf condition could also be weakened so that for a cover
of
, the sheaves
would glue on overlaps
only by quasi-isomorphism.
Unfortunately, over characteristic p, differential graded algebras work poorly for homotopy theory, due to the fact